In the acquisition of seismic data, seismic waves are used to interpret subsurface geological formations. Seismic prospecting employs a seismic source used to generate a wave that propagates through the earth to be detected by a seismic detector or receiver.
There is an inherent problem with data acquisition for a receiver in motion. Data acquired at later times need to be skewed spatially to match the zero-time reference. The schematic in FIG. 1 illustrates this concept. The vessel motion is towards the left and the receivers being towed are shown at two different times. The first time, T, is at the moment of the source or “shot,” this is T=0. The figure shows sources behind and in front of the cable. Source position is not relevant for this example. At time T=1, the cable has moved to the left and the trailing receivers have now moved up a full receiver interval, as compared to the receivers at T=0. To compensate for the inconsistent spatial positioning, the data at T=1 must be skewed spatially in the direction of boat movement. For this shot, all the data at time slice T=1 are skewed one receiver increment, Δx, in the direction of boat movement. The value of Δx is simply the product of boat speed and recording time:Δx=−VboatT.  (1)
The negative value for Δx implies that, for conventional geometries, where the source leads the cables, the data are moved to smaller offsets. Here, the data at T=1, for the receiver trace labeled 5, is mapped to location 4. This constant shift (mapping) is performed along time slices, thus preserving the original travel time of the data.
FIGS. 2 and 3 illustrate such mapping on shot-record coordinates (time versus offset). FIG. 2 shows a shot record and the curved lines represent the trajectory of reflected events. The horizontal dashed line corresponds to an arbitrary time slice prior to spatial skewing. FIG. 3 shows the data after spatial skewing. The originally vertical lines (FIG. 2) corresponding to the original trace positions are shown, now tilted, to illustrate the time-variant nature of the shift. A critical issue illustrated in FIG. 3 is that the steep trajectory of the seismic data along a shot record (curved lines) introduces a problem with aliasing when the time dip becomes large at greater offsets (data towards the right in FIG. 3). In this context, the term “aliasing” refers to the need to properly sample the data along the dashed, horizontal time slice indicated in FIG. 2. Sampling theory demands that the waveforms, horizontally sampled at the trace locations, must have at least two samples for the particular (spatial) frequency being sampled. If not, the sampled data are improperly sampled and are said to be aliased. When aliased, the frequency being sampled is incorrectly represented as a lower frequency (Sheriff, R. E., 1991, Encyclopedic Dictionary of Applied Geophysics, Fourth Edition: Soc. of Expl. Geophys.). Such aliasing problems are inherent in seismic data and represent a serious impediment to the routine implementation of the spatial-skew approach.
Thus, there is an interest in the art of seismic data processing to find improved methods to reduce the aliasing of seismic data while correcting for spatial-skew. The present invention addresses this interest.